update to solutions for tutorial 7

tutorials/tutorialS1_7-soln.ipynb

@@ -203,13 +203,126 @@
     "plt.show()"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "id": "a4a6f792-8aee-4208-9c9d-aa7a14a13d76",
+   "metadata": {},
+   "source": [
+    "To determine the mean concentration $\\bar{c}$ at the end of the pipe, we first need to determine the total amount of the solute in a small disk of length $\\Delta z$ at the end of the pipe.  The total amount of solute in the pipe is\n",
+    "$$\n",
+    "\\begin{align*}\n",
+    "\\Delta N &= \\int_0^R \\Delta z 2\\pi r dr\\, c(r)\n",
+    "\\end{align*}\n",
+    "$$\n",
+    "To get the mean concentration, we need to divide the total amount of solute in the disk by the volume of the disk, which is $\\Delta V = \\Delta z \\pi R^2$.  This gives\n",
+    "$$\n",
+    "\\begin{align*}\n",
+    "\\bar{c} &= \\frac{\\Delta N}{\\Delta V}\n",
+    "\\\\\n",
+    "&= \\frac{1}{\\Delta z \\pi R^2} \\int_0^R \\Delta z 2\\pi r dr\\, c(r)\n",
+    "\\\\\n",
+    "&= \\frac{2}{R^2} \\int_0^R dr\\, r c(r)\n",
+    "\\end{align*}\n",
+    "$$\n",
+    "\n",
+    "\n",
+    "To evaluate this integral, we can use the trapezoid rule code we developed in Q1 of tutorial 1 or the code from Q2 of tutorial 5.   "
+   ]
+  },
   {
    "cell_type": "code",
    "execution_count": null,
    "id": "d4ddc5f1",
    "metadata": {},
    "outputs": [],
-   "source": []
+   "source": [
+    "import numpy as np\n",
+    "import pylab as plt\n",
+    "\n",
+    "def v(r):\n",
+    "    v0 = 1.0e-2\n",
+    "    x = r/R\n",
+    "    return 2.0*v0*(1.0-x**2)\n",
+    "\n",
+    "def c(r):\n",
+    "    c0 = 0.2\n",
+    "    L = 1.0\n",
+    "    t = k*L/v(r)\n",
+    "    #print(t)\n",
+    "    return c0*np.exp(-t)\n",
+    "\n",
+    "\n",
+    "R = 1.0e-3\n",
+    "\n",
+    "\n",
+    "### Below is the code from Q2 of tutorial 5\n",
+    "### First we need to modify the function we are integrating:\n",
+    "def I_f(r):\n",
+    "    return r*c(r)\n",
+    "\n",
+    "N = 100+1\n",
+    "r_data = np.linspace(0, 0.9999*R, N)  # this needs to change to match our limits of integration\n",
+    "                                       # note that there is a problem exactly at r=R, so we will just\n",
+    "                                       # integrate a little bit below.\n",
+    "\n",
+    "I = 0.0\n",
+    "for i in range(N-1):\n",
+    "    ra = r_data[i]\n",
+    "    rb = r_data[i+1]\n",
+    "    I += 0.5*(rb-ra)*(I_f(ra)+I_f(rb))\n",
+    "\n",
+    "\n",
+    "print(I)\n",
+    "c_bar = 2/R**2 * I\n",
+    "print(f'c_bar = {c_bar:.5f} M')\n",
+    "\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a19612a1-4850-4662-aee2-bd172ac5b368",
+   "metadata": {},
+   "source": [
+    "We don't need to write the trapezoid rule code for ourselves.  There are many routines available on the internet that can do numerical integration for us.  One of them is the `quad` function from the `scipy` library."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "907e3901-b66c-4862-8aed-9881dffc1c3d",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import numpy as np\n",
+    "import pylab as plt\n",
+    "\n",
+    "def v(r):\n",
+    "    v0 = 1.0e-2\n",
+    "    x = r/R\n",
+    "    return 2.0*v0*(1.0-x**2)\n",
+    "\n",
+    "def c(r):\n",
+    "    c0 = 0.2\n",
+    "    L = 1.0\n",
+    "    t = k*L/v(r)\n",
+    "    #print(t)\n",
+    "    return c0*np.exp(-t)\n",
+    "\n",
+    "\n",
+    "\n",
+    "R = 1.0e-3\n",
+    "\n",
+    "\n",
+    "from scipy.integrate import quad\n",
+    "I = quad(I_f, 0, R)  # the output of quad is a tuple.  The first element is the value of the integral.  The\n",
+    "                     # second number is the estimated uncertainty of the value.\n",
+    "print(I)\n",
+    "c_bar = 2/R**2 * I[0]   # note: we use I[0], because we just want to use sthe value of the integral.\n",
+    "print(f'c_bar = {c_bar:.5f} M')\n",
+    "\n",
+    "\n",
+    "\n"
+   ]
   },
   {
    "cell_type": "markdown",
@@ -1289,7 +1402,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.9.8"
+   "version": "3.11.5"
   }
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  "nbformat": 4,